The tangent law applies specifically in two main contexts:

1. In trigonometry (Law of Tangents):

• The law of tangents relates the sides and angles of any triangle.
• It applies when you know either two sides and one angle or one side and two angles of a triangle.
• It is used to find unknown parts of a triangle.
• The formula is a−ba+b=tan⁡(A−B2)tan⁡(A+B2)\frac{a-b}{a+b}=\frac{\tan\left(\frac{A-B}{2}\right)}{\tan\left(\frac{A+B}{2}\right)}a+ba−b​=tan(2A+B​)tan(2A−B​)​, relating the difference and sum of two sides to the tangent of half the difference and half the sum of their opposite angles.
• The law of tangents has applications in various mathematical calculations involving triangles.

2. In magnetism (Tangent Law in Magnetism):

• This law applies to a magnetic needle or dipole placed in two perpendicular magnetic fields.
• The needle comes to equilibrium at an angle θ\theta θ to the magnetic meridian where tan⁡θ=BBH\tan \theta =\frac{B}{B_H}tanθ=BH​B​, with BBB the applied magnetic field at right angles to the horizontal component BHB_HBH​ of the Earth's magnetic field.
• This law is used to find the strength of unknown perpendicular magnetic fields.
• It specifically applies when the two magnetic fields are perpendicular to each other.

So, the tangent law applies in trigonometry when two sides and one angle, or one side and two angles are given in any triangle, and in magnetism when two perpendicular magnetic fields act on a magnetic needle or dipole.